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\begin{frontmatter}

\title{A mixed dynamic optimization with $\mu$-synthesis (D-K iterations) via gain scheduling for varying dynamics of decoupled twin-rotor MIMO system based on the method of inequality (MOI).\thanksref{footnoteinfo}} 
% Title, preferably not more than 10 words.

\thanks[footnoteinfo]{This research work is funded by the National Key R and D Program of China under Grant No. 2018YFB1702200 and National Natural Science Foundation (Grant No. 62076049). The content of this [report/study/article/publication] does not reflect the official opinion of the National Natural Science Foundation.}

\author[First]{Nadir Abbas} 
\author[Second]{Xiaodong Liu} 
%\author[Third]{Third C. Author}

\address[First]{Department of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, Liaoning, China 
    (e-mail: nadir10@mail.dlut.edu.cn, nadirabbas063@gmail.com)}
\address[Second]{Department of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, Liaoning, China 
    (e-mail: xdliuros@dlut.edu.cn)}


\begin{abstract}                % Abstract of not more than 250 words.
The complex structured system always demands efficient optimization to tackle all problems (constraints, parametric perturbation, multiobjective due to conflict of design), which must be achieved simultaneously.
This paper provides the experimental validation of dynamic robust mixed optimization based on gain scheduling for the varying constraints of a decoupled Twin Rotor MIMO System (TRMS). As mathematical models have unknown and unmatched nonlinear disturbances with un-modeled states during modeling, that need to be controlled to get optimize the response. Therefore, the design of a suitable controller for robust control of TRMS is a challenging task. To overcome this challenge, the controller design process is divided into two phases. The first phase adopts the Nonlinear Dynamic Inversion (NDI) based decoupled linearization process to obtain the Vertical Plane System (VPS) and Horizontal Plane System (HPS). To reduce the disturbance, the diagonal matrix and decoupling methods are applied simultaneously. In second phase, it covers the suitable choice of weighting functions (gain scheduling) which uses efficient order (reduced order) of the controller to provide satisfactory optimal response. The weighting functions are applied as the design parameters. In this process, the weights are selected in such a way to get the high gain for the low frequency and vice versa. The combined flexible approach with  the $\mu$-synthesis control as a mixed optimization make it mature algorithm to guarantee the robust stability and robust performance at extreme disturbance.
 The experimental validation of this control strategy verify the worth of the methodology due to optimal selection of the values of tuning parameters. For experimental implementation and validation of $\mu$-synthesis, a simulink/MATLAB coder is used.
\end{abstract}

\begin{keyword}
Perturbed MIMO system, robust stability performance, dynamic mixed optimization, gain scheduling, D-K iterations.
\end{keyword}

\end{frontmatter}
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\section{Introduction}
Applications about air vehicles caught interest due to their increasing control applications and flight control complexities. The modern control researchers are working on getting success in all such systems, which have some un-known non-linearities and un-modeled states of TRMS \cite{marconi2008aggressive,lara2010robustness}. TRMS is a type of Unmanned Air Vehicle (UAV).  Their ability to tilt their angle of flight, hovering, take-off, and landing in irregular locations provide special interest to researchers. A prototype of TRMS resembles to a helicopter which can be served as an effective tool for experiments in a real-time environment \cite{tastemirov2013complete}. Highly coupled, a higher degree of nonlinear dynamics, uncertainties, and gyroscopic torque needs to be tackled by an effective, robust controller. The control theory researchers are attracted towards problems due to its ongoing expanding applications.
Linear, nonlinear and intelligent control strategies are discussed as literature to understand the behavior of the TRMS as well as considered perturbations (noise, parametric) effect. Proportional Integral Derivative (PID) and Linear Quadratic Regulator (LQR) with output feedback control are linear strategies implemented 
 \cite{juang2008pid,wen2011twin}. Backstepping control strategy also implemented to understand the behavior of prototype in the presence of parametric uncertainties \cite{haruna2017dual}. Model Predictive Control(MPC) design also for TRMS presented with considering the matched, mismatched perturbations as disturbance \cite{raghavan2017practically}. Sliding Mode Control (SMC), Integral SMC and second-order SMC are implemented \cite{saroj2013sliding, young1999control, castanos2006analysis}. Chattering phenomena is the major disadvantage of the controller which may cause serious damage to the system. Adaptive second-order SMC also discussed \cite{liu2009single}. TRMS verifies that how important is to tackle this problem for the smooth output response \cite{yu2005sliding}. Experimental validation also represented by decentralized SMC strategy \cite{faris2017design}. Intelligent control like the PID-based fuzzy sliding mode control is considered as an excellent controller for such systems having some external disturbances mentioned in \cite{huang2013pid}. The neural networks control in \cite{pratap2010neural}, make sure the convergence of highly nonlinear system towards equilibrium. Adaptive control with the combination of intelligent and nonlinear control is also revised to understand the pattern of the controller as well as the system behavior. Adaptive neural networks backstepping control in \cite{liu2020adaptive} elaborated and adaptive fuzzy backstepping control discussed in \cite{liu2017adaptive}. Adaptive type-2 fuzzy backstepping control for the fractional-order nonlinear system also studied in \cite{jafari2019adaptive} to understand the worth of upcoming hot research in control. Controller design under some stability analysis discussed with coupling effect of a highly nonlinear system.
Nonlinear Dynamic Inversion (NDI) is a feedback linearization tool for TRMS used to reduce the complexity of a mathematical model of TRMS \cite{rahideh2012real}. Nonlinearities are canceled at any stability point by feedback linearization. The drawback of this method is that there may ignore some important nonlinearities, singularity, square matrix inversion. Large systems always required efficient modeling as well as numerical singularity avoidance \cite{bajodah2018aircraft, ansari2016generalized}. Dynamic inversion before controller implementation provide suitable environment to design the controller \cite{ansari2015generalized,ben2003generalized}. The coupled nonlinear system is decoupled to have simplified sub-systems named Vertical Plane System (VPS) and Horizontal Plane System (HPS).

Matched and mismatched uncertainties (perturbations) are also considered as un-known disturbances. The significant coupling effect must be focused on to manage the stability complexities of the system. The un-modeled states and parametric perturbations with coupling effect are tasks for any efficient controller to regulate the output of all the states. The complex structured system always demands efficient optimization to tackle all problems (constraints, parametric perturbation, multiobjective due to conflict of design), which must be achieved simultaneously \cite{whidborne1994robust,mihaly2021mu}. The contribution outline of the paper is enlisted as:

\begin{itemize}
    \item The weighting functions (tuning parameters) are used as a design parameters. It will combine with $\mu$-synthesis as a mixed optimization, to provide the combined benefits of optimization via MOI. 
	\item The diagonal matrix and decoupler are applied simultaneously (never implemented simultaneously) to reduce torque disturbance near to zero. In reward, efficient tracking control obtained.
	\item The flexible approach in term of design parameters make it mature algorithm to guarantee the robust stability and robust performance under extreme perturbations (external and internal disturbance).
	\item Real-time implementation under worse conditoions (noise and parametric variation provided to both rotors simultaniously with disturbance torque) validate the worth of robust optimization, which shows better response as compared to present research.
	\item Some important suggestions for control engineers are provided on the basis of experimental validation, to understand nature of control design as well as system behavior. 
	\end{itemize}
 The remaining of this paper have following sections as the mathematical modeling provided in section $2$. The NDI process for constrained dynamics of the TRMS and decoupled process discussed in section $3$. The section $4$ provide unstructured modeling with block diagram representation. Section $5$ elaborate the mixed optimization design procedure to validate the robust optimization. The controller design preliminaries with simulation response discussion elaborated in section $6$. Real-time setup description outline explained in section $7$ and conclusion based on validated results with some suggestions for control engineer presented in last section of the article which is $8$. 
 \section{Twin Rotor MIMO System (TRMS)}
Twin Rotor MIMO System (TRMS) is a prototype whose structure is almost near to a helicopter with limited degree of freedom. The modifications about such systems are required due to their wide applications in real life. The TRMS has two significant parts, the main rotor (vertical plane) and the tail rotor (horizontal plane). The main rotor with a higher diameter controls the movement of the beam on a vertical axis called pitch angle, while the tail rotor with a lower diameter covers the movement of the beam on a horizontal axis called yaw angle. The speed of the rotors manage equilibrium of the system. Each rotor of the TRMS connected with a separate DC supply motor, as shown in figure \ref{Twin-Rotor MIMO System}. 
\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{1 TRMS.png}
	\caption{Twin-Rotor Aerodynamic System.}\label{Twin-Rotor MIMO System}
\end{figure}
The rotational torque in the rotors of the TRMS produces cross-coupling torque to disturb the stability. This coupling effect is considered as the disturbance which is resolved by the decoupling process. The decoupling method is based on fixing one weight (motion) of the both rotors, and the system converted into two separate planes as VPS and HPS. 
\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{2 TRMS Block Diagram.png}
	\caption{Schematic Diagram of Twin-Rotor Aerodynamic System.}\label{Schematic Diagram of Twin-Rotor Aerodynamic System }
\end{figure}
Before understanding the mathematical modeling, we have to understand all varying parameters and required outputs of the TRMS. The TRMS is a lab apparatus that provides the understanding of the flight control of a helicopters \cite{tastemirov2013complete}. The considered system has two rotors as shown in figure \ref{Schematic Diagram of Twin-Rotor Aerodynamic System } and their design is most important because different forces are affecting the movement of the propellers. These forces are gravitational force, propulsive force, centrifugal force, frictional force and disturbance torque. To overcome the effects of these forces we provide control input voltage through motors. Understanding the mathematical assumptions, which are taken to understand and simplify the mathematical model. The rotor's dimensions are explained in figure \ref{Rotors dimensions with torque.} with their thrust directions:
 \begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{3 Rotors Dimensions and Torque Rotation.png}
	\caption{Rotors dimensions with torque .}\label{Rotors dimensions with torque.}
\end{figure}
 All nonlinear squared terms in the mathematical equations are linearized by the linearization process called NDI. The system movements are fixed along the horizontal plane and the azimuthal plane which are derived from the model \cite{young1999control}. The rotational movement of beam can be described as:
\begin{equation}\label{1}
\ {J_v}\frac{{{d^2}{\alpha _v}}}{{d{t^2}}} = {M_v}
\end{equation}
 where $ M_{v}$ considered as the whole momentum of the applied forces along the vertical plane, $ J_{v}$ shows the inertial momentum along the vertical plane axis. The parameter $ \alpha_{v}$ is required one output to control, called pitch angle (vertical axis). All the forces of the momentum can be summarized by the following representation of the momentum as: 
 
 \begin{equation}
 \ M_v = \ M_{{v_1}} + {M_{{v_2}}} + {M_{{v_3}}} + {M_{{v_4}}} + {M_{{v_5}}} + {M_{{v_d}}}
  \end{equation}
 The gravitational torque through the gravitational force is given as:
  \begin{equation}
 \ {M_{v1}} =  - {k_1}cos\left( {{a_v}} \right) - {k_2}sin\left( {{a_v}} \right)
  \end{equation}
 where $ k_{1}$ and $ k_{2}$ shows the constants which holds the mass mounted on the beam. The main propeller generates a momentum force which can be expressed through an equation as: 
 \begin{equation}
     \ {M_{{v_2}}} = {l_m}{F_v}\left( {{w_v}} \right)
 \end{equation}
 while $ l_{m}$ represents the length of beam,$ w_{v}$ shows the rotational velocity of the main propeller, and $ F_{v}(w_{v})$ shows the angular force of the main rotor. The moment force along vertical plane represented by the mathematical equation given below: 
 
 \begin{equation}
     \ {M_{({v_3})}} =  - {k_3} {\rm{\Omega }}_h^2 sin({a_v})cos({a_v}),
 \end{equation}
here, ${\Omega _h} = {\raise0.7ex\hbox{${{d_{\alpha_h}}}$} \!\mathord{\left/
 {\vphantom {{{d_{{\alpha _h}}}} {{d_t}}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${d_t}$}}$  is the beam velocity along vertical plane of the MIMO system, ${\alpha_h} $ considered as the yaw angle (rotation of the beam along azimuth plane), and $ k_{3}$   known as the constant coefficient. The frictional momentum depends upon the rotation of the beam with angular velocity in horizontal plane:
 \begin{equation}
     \ {M_{({v_4})}} =  - {k_{f_v}} {\Omega _v},
 \end{equation}
here ${\Omega _v} = {\raise0.7ex\hbox{${{d_{\alpha_v}}}$} \!\mathord{\left/
 {\vphantom {{{d_{{\alpha _v}}}} {{d_t}}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${d_t}$}}$  represents the angular velocity along the horizontal plane, while $k_f{_v}$ showing the constant quantity.
The momentum produced between the rotors due to applied input voltage (force) along horizontal axis:
\begin{equation}
     \ {M_{({v_5})}} =  - {k_{h_v}} {u _{h}},
 \end{equation}
where $u_{h}$ is the horizontal axis control input and $k_{hv}$ considered as the constant. The torque, which generate disturbance called the disturbance torque $M_{v_d}$ along vertical axis. The propeller force (propulsive force) along vertical axis (vertical plane) $F_v (w_v)$, produce the rotational velocity on the main rotor. The calculated velocity along the main rotor is given as:\\
\begin{multiline*}
     \widetilde {{F_v}} =  - 7.13 \times {10^{ - 19}}w_{v^5} - 3.79\times {10^{ - 16}}w_{v^4} + 2.41\times {10^{ - 11}}w_v^3 \\+ 1.87\times {10^{ - 8}}w_v^2 + 2.89\times {10^{ - 5}}{w_v} - 0.0124
\end{multiline*}\\
Summation of all the forces (torques) along horizontal axis can be calculated on vertical axis. The total torque produced by the tail rotor along horizontal axis produce a force with different spectrum (intensity of force). The torque (rotational effect of force) along horizontal axis can be calculated by a mathematical expression given below:  
\begin{equation}
     {J_h}({d^2}{a_h})/(d{t^2}) = {M_h}
\end{equation}
where $ M_{h}$ is the total momentum (force) along horizontal axis, $ J_h $ is total inertial force along vertical plane, ${J_h} = {k_4}cos2\left( {\alpha v} \right) + {k_5}$, the coefficients $k_4$ , $k_5$ are masses based constants of the beam. The total forces (momentum) along the horizontal axis represented by a mathematical expression:
\begin{equation}
  {M_h}=M_{h_1 }+M_{h_2 }+ M_{h_3 }+ M_{h_d},  
\end{equation}
All individual terms can be expressed as the propulsive momentum (force) along the tail rotor:
\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{4Thrust1.png} 
	\caption{4(a) Main rotor thrust and 4(b) Tail rotor thrust.}\label{4(a) Main rotor thrust and 4(b) Tail rotor thrust.}
\end{figure}
\begin{equation}
    {M_{{h_1}}} = {l_t}{F_h}\left( {{w_h}} \right)cos\left( {{a_v}} \right)
\end{equation}
where $l_t$ is beam length and $w_h$ is the rotational velocity, $F_h (w_h)$ is the propulsive momentum (force) of the rotor with angular velocity.
The frictional momentum based on the rotational velocity of the beam described as, 
\begin{equation}
    {M_{{h_2}}} =  - {k_{f_{h}}}{{\rm{\Omega }}_h}
\end{equation}
where $k_{f_{h}}$ is a constant.
The cross-sectional momentum due to control input action along the horizontal axis:
\begin{equation}
	{M_{{h_3}}} = {k_{v-{h}}}cos\left( {{\alpha _v}} \right){u_v}
\end{equation}
where $u_v$ is control input along the vertical axis, while $k_{v_h }$ is a coefficient constant. The torque along horizontal plane represented as $M_{h_d} $, called the disturbance torque due to tail rotor. The propeller force (propulsive force) along the horizontal axis (azimuthal plane) $F_v {w_v}$, produce the rotational velocity on the rotor. The calculated velocity along the tail rotor is given as:\\
\begin{multiline*}
     \widetilde {{F_h}} =  -  2.56 \times {10^{ - 20}}w_h^5 - 4.10\times {10^{ - 17}}w_h^4 + 3.17\times {10^{ - 12}}w_h^3 \\+ 7.34\times {10^{ - 9}}w_h^2 + 2.13\times {10^{ - 5}}{w_h} - 9.14,
\end{multiline*}\\
The equation of the main propeller (rotor) along vertical plane can be described as: 
\begin{equation}
    {I_v}(d{w_v})/dt = {u_v} - H_v^{ - 1}({w_v}),
\end{equation}
here $I_v$ is the inertial momentum of the main rotor (along vertical axis) and $w_v= H_v {u_h}$ shows the velocity (speed) of the main rotor known as the static velocity. The main rotor thrust represented in the figure \ref{4(a) Main rotor thrust and 4(b) Tail rotor thrust.}a and the velocity simulation results based on the experimental validation shown in figure \ref{4(a) Main rotor thrust and 4(b) Tail rotor thrust.}b. The 7th order equation for the vertical plane velocity of the main rotor is given below:
\begin{equation*}
    \widetilde {{w_v}} =  - 6.17 \times {10^3}u_v^7 - 1.30 \times {10^2}u_v^6 + 1.37 \times {10^4}u_v^5 + 1.50 \times {10^2}u_v^4 - 1.10 \times {10^4}u_v^3 - 3.76 \times {10^1}u_v^2 + 7.33 \times {10^3}{u_v} - 5.36
\end{equation*}
The tail rotor motion (speed) along horizontal plane can be represented by a mathematical equation as:

\begin{equation}
    {I_h}(d{w_h})/dt = {u_h} - H_h^{ - 1}({w_h}),
\end{equation}

here $I_h$ is inertial momentum of tail rotor (along horizontal axis) and $w_h= H_h {u_h}$ shows the velocity (speed) of the main rotor known as the static velocity.
The tail rotor thrust and the velocity simulation results based on the experimental validation. The 5th order equation of the rotational velocity given below:\\
\begin{multiline*}
    \widetilde {{w_h}} =  - 6.17 \times {10^3}u_h^5 - 1.30 \times {10^2}u_h^4 + 1.37 \times {10^4}u_h^3 \\+ 1.50 \times {10^2}u_h^2 - 1.10 \times {10^4}{u_h} - 37.6
\end{multiline*}

\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{5Velocity1.png}   
	\caption{5(a) Main rotor velocity and 5(b) Tail rotor velocity.}\label{ 5(a) Main rotor velocity and 5(b) Tail rotor velocity.}
\end{figure}


By rearranging both equations (15) and (16) together:

\begin{equation}
    \frac{{d{\alpha _v}}}{{dt}} = {{\rm{\Omega }}_v}
\end{equation}
\begin{equation}
    \frac{{d{\alpha _h}}}{{dt}} = {{\rm{\Omega }}_h}
\end{equation}
The state space model of the TRMS (6th order nonlinear system) represented in the mathematical modeling with the control input voltages $u_h$ (horizontal plane or yaw angle) and $u_v$ (vertical plane or pitch angle). The output angles are yaw (azimuth) angle $\alpha_h$ and pitch (vertical) angle $ \alpha_v$. 
To understand the full dynamic response of the system, it must be categorized as the multivariable system with the highly nonlinear behavior. The TRMS model having two channels which can never be considered as the independent channels. This cross-coupled property is known as the coupling effect. The coupling effect must be countered by decoupling procedure and converted it into the independent two-channel system. The system coefficients values of the model enlisted in table \ref{table:2}. 


\begin{table}[h]
\begin{center}
\begin{minipage}{174pt}
\caption{Parameters of TRMS.\cite{cheng2018cascade}}\label{tab1}%
\begin{tabular}{@{}llll@{}}
\toprule
Symbol & Description & Unit \\
\midrule
 $I_v$  & 1/6100    & $kgm^2$\\
 $I_h$ &  1/37000  & $kgm^2$   \\
 $J_v$ & $3.00581 \times {10^{ - 2}}$ & $kgm^2$\\
 $k_1$    & $5.00576 \times {10^{ - 2}}$ & $Nm$\\
 $k_2$    &   $9.0036 \times {10^{ - 2}}$   & $Nm$\\
 $k_3$    & $2.12485 \times {10^{ - 2}}$   & $Nm{s^2}/ra{d^2}$   \\
 $k_4$    & $2.3790412485 \times {10^{ - 2}}$    & $kgm^2$\\
 $k_5$    & $3.00962 \times {10^{ - 3}}$    & $kgm^2$\\
 $k_{f_h}$    & $5.88996 \times {10^{ - 3}}$    & $Nm-s/rad$\\
 $k{f_v}$    & $1.27095 \times {10^{ - 2}}$    & $Nm-s/rad$\\
 $k_{h_v}$    & $4.17495 \times {10^{ - 3}}$    & $Nm$\\
 $k_{v_h}$    & $-1.7820 \times {10^{ - 2}}$    & $Nm$\\
 $l_m$    & $0.202$    & $m$\\
 $l_t$    & $0.216$    & $m$\\ 
\botrule
\end{tabular}

\end{minipage}
\end{center}
\end{table}

\section {Nonlinear Dynamic Inversion (NDI) and Decoupling}
Feedback linearization control is also known as NDI control which establishes a supporting platform for linear control. The basic idea to embed this strategy is the cancellation of nonlinear terms as well as having a simplified mathematical model. Different variables about the angular position described in a given form:  

${\alpha _v} = {\alpha _{v,nom}} + \delta {\alpha _v},\;\;{w_v} = {w_{v,nom}} + \delta {w_v},
\;\;\;
{{\rm{\Omega }}_v} = {{\rm{\Omega }}_{v,nom}} + \delta {{\rm{\Omega }}_v}$

${\alpha _h} = {\alpha _{h,nom}} + \delta {\alpha _h},\;\;\;{w_h} = {w_{h,nom}} + \delta {w_h},\;\;{{\rm{\Omega }}_h} = {{\rm{\Omega }}_{h,nom}} + \delta {{\rm{\Omega }}_h}$

where  $\alpha _{v,nom}, w_{v,nom}, {{\rm{\Omega }}_{v,nom}}, 
{\alpha _{h,nom}}, {w_{h,nom}}, {{\rm{\Omega }}_{h,nom}}$ represents the corresponding values and $\delta {\alpha _v},\delta {w_v} , \delta {_{\rm{\Omega }}{_v}}, \delta {\alpha _h},\delta {w_h},\\ \delta {_{\rm{\Omega }}{_h}}$ shows deviations from nominal values. The motor voltages are shown as
${u_v} = {u_{v,nom}} + \delta {u_v},{u_h} = {u_{h,nom}} + \delta {u_h},\;$
All the output states are supposed to be converged at the origin,
${{\rm{\Omega }}_{h,nom}} = 0, {{\rm{\Omega }}_{v,nom}} = 0.$
After some basic mathematical operations and assumptions, given in \cite{cheng2018cascade}, linearized system equations are obtained here: 
\begin{equation}
    {I_v}\frac{{d\delta {w_v}}}{{dt}} = \delta {u_v} - \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/
 {\vphantom {1 {{k_{{H_v}}}}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${{k_{{H_v}}}}$}}} \right)\delta {w_v}
\end{equation}
\begin{equation}
    {I_h}\frac{{d\delta {w_h}}}{{dt}} = \delta {u_h} - \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/
 {\vphantom {1 {{k_{{H_h}}}}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${{k_{{H_h}}}}$}}} \right)\delta {w_h}
\end{equation}

\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{8 Linearized model block diagram.png}     
	\caption{Open loop block diagram of TRMS . }\label{Open loop block diagram of TRMS .}
\end{figure}

The block diagram of the TRMS system with the coupling effect is represented in figure \ref{Open loop block diagram of TRMS .}. The cross-coupling effect with all other unwanted perturbations makes the TRMS behavior very complex. The horizontal plane angle can be fixed by posing the value of  $u_h=0$. Decoupling makes the system into subsystems as VPS and HPS \cite{loutfi2019real}. The transfer function of both subsystems is given below:
\begin{equation}
    {G_v}\left( s \right) = \frac{{111.2}}{{0.390{s^3} + 0.3835{s^2} + 1.454s + 1}}
\end{equation}
\begin{equation}
    {G_h}\left( s \right) = \frac{{111.2}}{{5.64{s^2} + 3.97s + 1}}
\end{equation}
Subsystems are obtained by putting second control input equal to zero. The NDI can be affected by the singularity during the inversion process. The rank of the system matrix will be changed that generates a discontinuous behavior, which tends to go elements in matrix unbounded. Such kind of drawbacks can be covered by augmentation of scaling factor in \cite{ansari2016guidance}, elaborated as: 
\begin{equation}
    \dot v\left( t \right) =  - v\left( t \right) + \frac{\gamma }{{{e_z}{{\left( t \right)}^2}}},\;v\left( 0 \right) > 0
\end{equation}
where, $ \gamma $ is constant while ${e_z}{{\left( t \right)}}$ represents the tracking of pitch angle and yaw angle. A negative sign represents the convergence towards origin and asymptotic stability confined in equation 24 with tracking control of the angles in the above expression. 

\section {Unstructured Modeling}
The varying structure type systems always have some unknown states and mathematical complexities during modeling. The description of uncertain parameters is represented in mathematical expressions and notations. We suppose that $J_h$ is inertial momentum along the horizontal axis and $k_{F_h }, k_{F_v }$ are coefficients of the generated thrust of rotors. Rotors have some velocity gains $k_{H_h }, k_{H_v}$. The coefficients, $k_{f_h }, k_{f_v }$ and $k_{v_h }, k_{h_v}$ are frictional momentum as well as cross momentum coefficients respectively. $R_V$ is returned torque (coupling effect) between rotors. All these $10$ modeled parameters have a dependency with our main two outputs named pitch angle and yaw angle. 
\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{7inputoutput1.png}     
	\caption{7(a) Block diagram of decoupled TRMS input/output and 7(b) Block diagram of coupled TRMS input/output .}\label{7(a) Block diagram of decoupled TRMS input/output and 7(b) Block diagram of coupled TRMS input/output .}
\end{figure}

In addition, we suppose that the inertial momentum $J_h$, with coefficients  $k_{F_h }, k_{F_v }, k_{H_h }, k_{H_v}$, have the error estimation up to 10\%. The remaining coefficients have the error estimation up to 5\%. The mathematical expressions of the TRMS represents the system behavior as the controlled plant:
\begin{equation}
    G = \left[ {\begin{array}{*{20}{c}}
{{G_v}}\\
{{G_h}}
\end{array}} \right]
\end{equation}
where,
\begin{equation}
    y = G\left[ {\begin{array}{*{20}{c}}
{{M_d}}\\
u
\end{array}} \right],
y = \left[ {\begin{array}{*{20}{c}}
{{\alpha _h}}\\
{{\alpha _v}}
\end{array}} \right],
u = \left[ {\begin{array}{*{20}{c}}
{{u_h}}\\
{{u_v}}
\end{array}} \right],
{M_d} = \left[ {\begin{array}{*{20}{c}}
{{M_{{d_h}}}}\\
{{M_{{d_v}}}}
\end{array}} \right]
\end{equation}
The system schematic model of the TRMS with their input-output connections are represented in the figure \ref{7(a) Block diagram of decoupled TRMS input/output and 7(b) Block diagram of coupled TRMS input/output .}a and figure \ref{7(a) Block diagram of decoupled TRMS input/output and 7(b) Block diagram of coupled TRMS input/output .}b. Now, the basic mathematical description of the uncertain model discussed below. 
\begin{figure}[t!]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{11 SINGULAR VALUE OF TRMS.png}  
	\caption{Singular value of TRMS}\label{Singular value of TRMS. }
\end{figure}
Let us introduce the representation,
$G = \left[ {\begin{array}{*{20}{c}}
{{G_d}}&{{G_u}}
\end{array}} \right] while,\\ \;\;$
${G_d} = \left[ {\begin{array}{*{20}{c}}
{{G_{{d_h}}}}&{{G_{{d_v}}}}
\end{array}} \right]\;\;\;\;\;\;\;\;\;$
${G_u} = \left[ {\begin{array}{*{20}{c}}
{{G_{{u_h}}}}&{{G_{{u_v}}}}
\end{array}} \right]$

such that 
\begin{equation}
    y = {G_d}{M_d} + {G_u}u
\end{equation}
According to the above expression, $G_d$ represents the disturbance of the plant as matrix and $G_u$ shows the transfer matrix of the control signal. The singular value of the un-certain plant (nonlinear plant) is represented in the figure \ref{Singular value of TRMS. }, which is the frequency response of the system. The un-certain plant has some basic requirements in the presence of perturbations (internal and external disturbance): 
\begin{equation}
    u = \left[ {\begin{array}{*{20}{c}}
{{K_r}}&{{K_y}}
\end{array}} \right]{\left[ {\begin{array}{*{20}{c}}
r&{ - {y_c}}
\end{array}} \right]^T} = {K_r}r - {K_y}{y_c}
\end{equation}
where $K_y$ represents the feedback matrix function and $K_r$ is the transfer function matrix of the pre-filter. The closed-loop model (uncertain TRMS) shown in figure \ref{9(a) Robust stability 9(b) Robust Performance.}a, represents the controller stability response, performance requirement, and disturbance matrix of the noise function. While figure \ref{9(a) Robust stability 9(b) Robust Performance.}b represents the robust performance.
 \section{Mixed optimization with robust performance validation}
The gain scheduling actually depended on the choice of weighting functions which are chosen by considering the open-loop response of the weighted plant, so effectively the weights $W_p$ and $W_u$ are the design parameters. This means that the design problem can be formulated as in the method of inequalities, with the parameters of the weighting functions used as the design parameters to satisfy the set of closed-loop performance inequalities. Such an approach to the MOI overcomes
the limitations of the MOI. The designer does not have to choose the order of the controller, but instead chooses the order of the weighting functions. With low-order
weighting functions, high-order controllers can be synthesized that often lead to significantly better performance or robustness than if simple low-order controllers were
used. The design problem is now stated as follows.
Design Procedure 
1. Define the plant $G_v, G_h$ and define the functional.
2. Define the values of $e_y and e_z$.
3. Define the form and order of the weighting functions $W_p$ and $W_u$. Bounds should
be placed to ensure that $W_p$ and $W_u$ are stable and minimum phase to prevent undesirable pole/zero cancellations. The order of the weighting functions, and hence the value should be small initially.
4. Define initial values of $W_p$ based on the open-loop frequency response of the
plant.
5. Implement the MBP, or other appropriate algorithms to find a $(W_p,W_u)$ that satisfies inequalities. If a solution
is found, the design is satisfactory; otherwise, either increase the order of the
weighting functions, or relax one or more of the desired bounds, or try again.
6. With satisfactory weighting functions $W_p$ and $W_u$, a satisfactory feedback controller
is obtained.Different variables like $"r" , "d"$, and $"n"$ represent the reference input, input disturbance and noise respectively. Output angles as yaw angle $\alpha_h$ and pitch angle  $\alpha_v$ are required to control (measure) under all kinds of perturbations (noise, parametric, coupling effect ). Output tracking control signals $e_y$ and $e_u$ are error tracking signals. The output feedback vector $y_c=y+W_{nn}$, is the vector-matrix having measured noise $n$ and $W_n$  filter for the noise shaping.
Following weighted functions of system  required error tracking output $(e_y\;\; and\;\; e_u)$ equation must satisfy the condition:\\ 
\begin{multiline*}
    \left[ {\begin{array}{*{20}{c}}
{{e_y}}\\
{{e_u}}
\end{array}} \right] =\\ \left[ {\begin{array}{*{10}{c}}
{{W_p}\left( {{S_o}{G_u}{K_r} - M} \right)}&{{W_p}{S_o}{G_d}}&{ - {W_p}{S_o}{G_u}{K_y}{W_n}}\\
{{W_u}{S_i}{K_r}}&{ - {W_u}{S_i}{K_y}{G_d}}&{ - {W_u}{S_i}{K_y}{W_n}}
\end{array}} \right]\left[ {\begin{array}{*{10}{c}}
{\begin{array}{*{10}{c}}
r\\
d
\end{array}}\\
n
\end{array}} \right]
\end{multiline*}\\
while $S_i=(I+K_y G_u )-1$ and $S_o=(I+G_u K_y )-1$ shows input, output sensitivity matrix function respectively.\\  
\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{12Robustness1.png} 
	\caption{9(a) Robust stability 9(b) Robust Performance.}\label{9(a) Robust stability 9(b) Robust Performance.}
\end{figure}


\begin{table}[h]
\begin{center}
\begin{minipage}{174pt}
\caption{Weighted Functions.}\label{tab1}%
\begin{tabular}{@{}llll@{}}
\toprule
Functions & Description \\
\midrule
  $W_p (S_o G_u K_r-M)$   & Weighting difference\\
  $W_p S_o G_d$  & Weighted sensitivity to disturbance   \\
  ${W_p}{S_o}{G_u}{K_y}{W_n}$ & Weighted sensitivity to noise\\
  $W_u S_i K_r$ & Weighted control action due to reference\\
$W_u S_i K_y G_d$   & Weighted control action due to disturbance\\
 $W_u S_i K_y W_n$   & Weighted control action due to noise   \\
\botrule
\end{tabular}

\end{minipage}
\end{center}
\end{table}

The performance criterion requires the transfer function matrix from the exogenous input signals $r,d$  and $n$ to the output signals $e_y$ and $e_u$  to be small, for all the possible output of uncertain plant model $G$. The transfer function matrices $W_p$ and $W_u$ are used to reflect the relative importance of different frequency ranges for which the performance requirements should be fulfilled. The transfer function matrices which constitute the transfer function matrix between the inputs and outputs of the extended system are described in table \ref{table:3} . The controller design task is to regulate required output: 
\begin{equation}
    K = \left[ {{K_r}\;\;\;\;\;\;\;\;\;\;\;\;\;\;{K_y}} \right]
\end{equation}
That must elaborate and satisfied the enlisted properties under perturbations \cite{callier2012linear}. Robust stability under perturbations must meet the required response by satisfying closed-loop nominal performance and robust response conditions as mentioned \cite{jastrzebski2011discussion,morari1989robust}. The condition for nominal performance:\\
\begin{multiline*}
    {\left[ {\begin{array}{*{20}{c}}
{{W_p}\left( {{S_{o,nom}}{G_{u,nom}}{K_r} - M} \right)}\\&{{W_p}{S_{o,nom}}{G_{d,nom}}}\\&{ - {W_p}{S_{o,nom}}{G_{u,nom}}{K_y}{W_n}}\\
{{W_u}{S_{i,nom}}{K_r}}\\&{ - {W_u}{S_{i,nom}}{K_y}{G_{d,nom}}}\\&{ - {W_u}{S_{i,nom}}{K_y}{W_n}}
\end{array}} \right]_\infty }\\ < 1\\
\end{multiline*}
The condition for robust performance:
\begin{equation}
    {\left[ {\begin{array}{*{20}{c}}
{{W_p}\left( {{S_o}{G_u}{K_r} - M} \right)}&{{W_p}{S_o}{G_d}}&{ - {W_p}{S_o}{G_u}{K_y}{W_n}}\\
{{W_u}{S_i}{K_r}}&{ - {W_u}{S_i}{K_y}{G_d}}&{ - {W_u}{S_i}{K_y}{W_n}}
\end{array}} \right]_\infty } < 1
\end{equation}
Above conditions must be satisfied for $G$.
\section{Controller Design Preliminaries}
The basic controller design with their attributes is covered in this section.  
This section elaborates the appropriate system's controller with all possible weighting functions. The structured and unstructured perturbations with nominal robust performance requirements are achieved by a robust optimization strategy.
The controller design of a higher-order variable structure system is a very important and extremely complicated task. Stability of complex system needs concentration towards system stability analysis as discussed \cite{doyle1982analysis, ng1988interactive}. Quality of work and the increasing response of production, make it more popular.  The mathematical model of any VSS type system represents the dynamics. The MIMO systems like TRMS are higher-order systems with highly nonlinear behavior. 
%%%%%%%%%%%%%%%%%%%%%%%%%
A mixed optimization method based on MOI provide flexibility in term of controller design specifications.  
For example, the controller for system may be required
to have a rise time of less than one second, a settling time of less than five seconds
and an overshoot of less than 10 $percent$. In such cases, it is obviously more logical and
convenient if the design problem is expressed explicitly in terms of such inequalities.
The method of inequalities \cite{zakian1973design} is a computer-aided multiobjective design
approach, where the desired performance is represented by such a set of algebraic
inequalities and where the aim of the design is to simultaneously satisfy these inequalities.


%%%%%%%%%%%%%%%%%%%%%



$\mu$-synthesis considered as the robust control strategy which mitigates the perturbations to get required results.  Outline for the design of strategy contain two steps, first one is to derive robust performance in the presence of structured and unstructured perturbations which would be transformed towards stabilization. The next step is about the design specifications of the iterative method \cite{gu2005robust}. $D-K$ iteration based $\mu$-synthesis method as shown in the figure \ref{Block diagram of controller }, based on weighting functions to ensure the credibility of control strategy \cite{slavov2013real,doyle1985structured}. The controller design contain four major steps which are being elaborated in figue \ref{Controller design flow chart.}. The robust control toolbox is used to verify the functions property $"dksyn"$.The basic idea of $D-K$ iteration
is to find an optimal controller $(K)$ and an optimal
scaling $(D)$ in an iterative way, Their aspects are related (required) to reduce the cost value of the singular value function.
\begin{figure}[t!]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{13 Controller Block Diagram.png}  
	\caption{Block diagram of controller}\label{Block diagram of controller }
\end{figure}
The function, ${P_d}\left( z \right) = {F_U}\left( {{N_d},\delta} \right)$ , represents the transfer function of discrete open loop TRMS. 

\begin{figure}[!h]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{14 DK Iterationssteps.png}  
	\caption{Controller design flow chart.}\label{Controller design flow chart.}
\end{figure}

The block structure of the $\Delta {P_d}$ can be elaborated here by a mathematical equation given below as: 
\begin{equation}
    \Delta {P_d}: = \left\{ {\left[ {\begin{array}{*{20}{c}}
\Delta &0\\
0&{{\Delta _F}}
\end{array}} \right]:\Delta  \in {{\cal R}^{10 \times 10}},\;{\Delta _F} \in {C^{6 \times 4}}} \right\}
\end{equation}
The parametric uncertainties as a block, are being notified by $\Delta {P_d}$ and $\Delta $ in the mathematical expression. The fictitious perturbation block is $\Delta_F$, contain robust performance for $\mu-$approach. The controller task is to get the discrete stabilizing controller gain $K_d$, which must satisfy the condition:
\begin{equation}
    {\mu _{{\Delta _{{P_d}}}}}\left[ {{F_L}\left( {{N_d},{K_d}} \right)\left( {j\omega } \right)} \right] < 1
\end{equation}
where $F_L (N_d,K_d )$ is the transfer matrix of the closed-loop dynamic system. The robust based performance of the system must have limited output which is less than one, i.e.
\begin{equation}
    {F_U}{\left[ {{F_L}\left( {{N_d},{K_d}} \right),\Delta {P_d}} \right]_\infty } < 1
\end{equation}
The decoupled system can never be purely decoupled during real time implementation. The ideal model is selected as a diagonal matrix to reduce the coupling effect of the system as,  
\begin{equation}
    M\left( s \right) = \left[ {\begin{array}{*{20}{c}}
{{\omega _{m1}}}&0\\
0&{{\omega _{m2}}}
\end{array}} \right]
\end{equation}

\begin{figure}[t!]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.7\linewidth]{11inverse performance of model1.png} 
	\caption{12(a) Inverse weighting function 12(b) Inverse performance 12(c) Model Frequency 12(d) Sensor noise}\label{12(a) Inverse weighting function 12(b) Inverse performance 12(c) Model Frequency 12(d) Sensor noise }
\end{figure}

\begin{equation}
    {\omega _{m1}} = \frac{1}{{1.2{s^2} + 1.1s + 1}}
\end{equation}

\begin{equation}
    {\omega _{m2}} = \frac{1}{{1.8{s^2} + 1.5s + 1}}
\end{equation}
The magnitude response of the azimuth angle magnitude response is faster than the pitch angle magnitude response. Several iterations are performed, several times to get the optimal response under all nonlinearities (parametric, modeling error, noise signal) of the system \cite{cheng2018cascade}. The weighting functions are known as tuning parameters to get the best performance. The experimental evidence based weighting (tuning parameters) functions are defined here:
\begin{equation}
    {W_P}\left( s \right) = \left[ {\begin{array}{*{20}{c}}
{8.5 \times {{10}^{ - 2}}\frac{{80s + 1}}{{80s + {{10}^{ - 3}}}}}&{ - 0.03}\\
{0.02}&{7.0 \times {{10}^{ - 1}}\frac{{501s + 1}}{{501s + {{10}^{ - 3}}}}}
\end{array}} \right]
\end{equation}
The weighting (tuning parameters) function for input control signal given below:
\begin{equation}
    {W_u}\left( s \right) = \left[ {\begin{array}{*{20}{c}}
{4.1 \times {{10}^{ - 5}}\frac{{0.05s + 1}}{{{{10}^{ - 4}}s + 1}}}&0\\
0&{2.306 \times {{10}^{ - 4}}\frac{{0.1s + 1}}{{{{10}^{ - 4}}s + {{10}^{ - 3}}}}}
\end{array}} \right]
\end{equation}
The tuning parameters (weighting function) for both angles are chosen to limit the output variation within ${\rm{[ - 0}}{\rm{.8, 0}}{\rm{.8], [ - 0}}{\rm{.5, 1]}}$ for the azimuth angle and the pitch angle respectively.

The inverse performance functions, inverse control functions are represented in the figure \ref{12(a) Inverse weighting function 12(b) Inverse performance 12(c) Model Frequency 12(d) Sensor noise }a and figure \ref{12(a) Inverse weighting function 12(b) Inverse performance 12(c) Model Frequency 12(d) Sensor noise }b respectively. The weighting functions (tuning parameters) are selected on the basis of required constraints. The closed-loop system response on the basis of experimental evidences is very sensitive. Large number of the experiments required to get best weighting function value. Precise tuning for the weighting function is the key point for the optimal robust response. The transfer function of noise matrix is given below as:
\begin{equation}
    {W_n}\left( s \right) = \left[ {\begin{array}{*{20}{c}}
{{\omega _n}\left( s \right)}&0\\
0&{{\omega _n}\left( s \right)}
\end{array}} \right]
\end{equation}
where the noise transfer function matrix ${\omega _n} = {10^{ - 2}}\frac{s}{{s + 1}}$ worked as high pass filter with the significant output more than 10 $rad/s$. The noise matrix signal response shown in the figure \ref{12(a) Inverse weighting function 12(b) Inverse performance 12(c) Model Frequency 12(d) Sensor noise}d.  The D–K iterations are shown in table \ref{table:4}.\\

\begin{table}[h]
\begin{center}
\begin{minipage}{174pt}
\caption{D-K Iterations.}\label{tab1}%
\begin{tabular}{@{}llll@{}}
\toprule
Iterations & Controller order & value of $\mu$\\
\midrule
  1   &16& 232.315\\
  2  & 16 & 4.942 \\
  3 & 20 & 1.336\\
  4 & 22 & 0.986\\
  5   & 24 & 0.969 \\
\botrule
\end{tabular}

\end{minipage}
\end{center}
\end{table}




Limited iterations are performed as mentioned in the above table, that reduces the cost value up to 0.968 with 24th order of controller. The decoupled system response based on robust stability as well as robust performance is validated in figure \ref{13(a) Robust Stability, 13(a) Robust performance}a and figure \ref{13(a) Robust Stability, 13(a) Robust performance}b respectively.
The system output can be disturbed by noises at frequency ranges from 5 $rad/s$ to 10 $rad/s$. The decoupled TRMS output response shows that this effect is negligible due to the decoupler and diagonal matrix. The step input is provided to express the decoupled system response under perturbations (structured and unstructured).
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{12Robustness1.png}   
	\caption{13(a) Robust Stability, 13(a) Robust performance}\label{13(a) Robust Stability, 13(a) Robust performance}
\end{figure}

Pitch angle as well as yaw angle with their control input response elaborated in figure \ref{14(a) Pitch Angle With Control Action, 14(b) Yaw Angle With Control Action }a and figure \ref{14(a) Pitch Angle With Control Action, 14(b) Yaw Angle With Control Action}b respectively. 
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{13angle response1.png}
	\caption{14(a) Pitch Angle With Control Action, 14(b) Yaw Angle With Control Action}\label{14(a) Pitch Angle With Control Action, 14(b) Yaw Angle With Control Action}
\end{figure}

\section{Experimental Setup With System Internal Structure}
In this section, we elaborate on the key concept of real-time implementation and system interconnections through system integrated circuits. The structure of the open-loop system with their all input-output $(8/8)$ ports is represented in figure \ref{System Internal Structure of Open Loop TRMS }. The internal structure of the system is also labeled with ports, to understand implementation more precisely for the reader. The open-loop system block diagram with their reference input points and required output ports is represented in figure \ref{Open Loop Block Diagram With Input/Output Scheme }.

\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{23 System internal structure of open loop TRMS.png}  
	\caption{System Internal Structure of Open Loop TRMS}\label{System Internal Structure of Open Loop TRMS }
\end{figure}
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{24 Open loop block diagram.png}   
	\caption{Open Loop Block Diagram With Input/Output Scheme}\label{Open Loop Block Diagram With Input/Output Scheme }
\end{figure}
The schematic diagram of the closed-loop system with important variables description elaborated in figure \ref{System Internal Structure of Closed Loop TRMS } and number of input-output ports also provided to understand internal structure easily. The block diagram of closed-loop system interconnections with a short description elaborated in figure \ref{System Internal Structure of Closed Loop TRMS } and direction of the arrow showing signal flow of TRMS in figure \ref{Closed Loop Block Diagram With Input/Output Scheme }.
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{25 System internal structure of closed loop TRMS.png}    
	\caption{System Internal Structure of Closed Loop TRMS}\label{System Internal Structure of Closed Loop TRMS }
\end{figure}
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{26 Closed loop scheme.png}     
	\caption{Closed Loop Block Diagram With Input/Output Scheme}\label{Closed Loop Block Diagram With Input/Output Scheme }
\end{figure}

The real-time implementation with the help of a robust designed controller validates the controller worth under disturbances (noise signal, un-modeled states, parametric, coupling effect). Experimental setup connected with personal computer represented in figure \ref{Experimental Setup With Prototype. }, implemented for closed-loop system via built-in drive interface. 
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{27 TRMS NEW PROTOTYPE.png} 
	\caption{Experimental Setup With Prototype.}\label{Experimental Setup With Prototype. }
\end{figure}
\begin{figure}[!t]
	\centering
	%width=9cm,height=5cm
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{experimental setup.png} 
	\caption{Flow Chart TRMS Laboratory Setup Implementation.}\label{Flow Chart TRMS Laboratory Setup Implementation. }
\end{figure}

The experimental processing setup of TRMS with all required steps are mentioned in figure \ref{Flow Chart TRMS Laboratory Setup Implementation. } to understand the laboratory hardware prototype implementation. The reference speed (velocity) variations provided to azimuthal angle (tail rotor) and pitch angle (main motor) with limited amplitude are represented in figure \ref{Rotors Speed (rpm). }. The limited varying speed provided to validate the system robust response with stability credibility of the controller. The experimental output response of the pitch angle and yaw angle with their control action shows in figure \ref{Experimental Response of TRMS . } and figure \ref{Experimental Response of Control Action . } that validates the system's sharp response towards convergence within a limited variation range of voltage. 	
Experimental results describe that linearized systems have the almost same response as compared to nonlinear system responses. A high level of noise (disturbance), causes a serious problem with the actuators and input control signal as an error. To get the actual actuator input the first-order filter based on the Butterworth filter used.
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{29 RPM OF ANGLES.png}  
	\caption{Rotors Speed (rpm).}\label{Rotors Speed (rpm). }
\end{figure}
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{30 EXPERIMENTAL PITCH YAW ANGLE.png}  
	\caption{Experimental Response of TRMS .}\label{Experimental Response of TRMS . }
\end{figure}
\begin{figure}[!t]
	\centering
	%Requires \usepackage{graphicx}
	\includegraphics[width=0.9\linewidth]{31 EXPERIMENTAL CONTROL INPUT.png} 
	\caption{Experimental Response of Control Action .}\label{Experimental Response of Control Action . }
\end{figure}


\section{Conclusion}

The work reported in this paper is the outcome of several attempts to design the robust controller and robust performance for the coupled TRMS, which is a prototype model of a helicopter. This model is a higher order system having a significant coupling effect between the main (pitch) rotor and tail (yaw) rotor. The design of a suitable controller for the robust control of the TRMS is a challenging task. Due to nonlinearities in the system, some assumptions have been made while deriving its mathematical model. The dynamic inversion process reduces the complexity behind the mathematical model assumptions and stability analysis (stability performance and robust performance) provide satisfactory response to design a robust optimization. The weighting functions (gain scheduling) used as design parameters, which have been selected for the robust control of varying constraints of the system, so that high gains have been achieved for the low frequency and low gains achieved for the high frequency. The weights have been selected iteratively through the stability and robustness performance simulation results behavior. The system output can be disturbed by noises at frequency ranges from 5 $rad/s$ to 10 $rad/s$. The decoupled TRMS output response shows that this effect is negligible due to the decoupler and diagonal matrix. Under such conditions, the system must be controlled with an efficient sharp response, which is the objective of this research work. The real-time implementation validates the worth of the control strategy by converging output response in the presence of perturbations (noise signal, un-modeled states, parametric, coupling effect). The research experience about the controller design (D-K iterations) and real-time implementation verifies that the decoupled TRMS has the following suggestions for control engineers.
\begin{itemize}
	\item The modern control research for real-time implementation in this paper verifies that the controller must be of higher order ($n=24$). The higher-order controllers have the best performance for the highly nonlinear coupled system.
	\item Noise signal with high amplitude cause serious contamination for input actuators and high range frequency.
	\item The disturbance torque of tail rotor cannot reduced to zero through decoupling process in real-time performance.
	\item The controller report verifies the robust stability as well as robust performance to modeled perturbations (uncertainty). The maximum tolerance ability against perturbations is more than 500\%.
	\item There is no instability cause at the frequency 0.001 rad/s due to modeled perturbation.
	\item The system robust margin against performance is 1.051. The robust performance margin is o.968 in term of model perturbation exist with a size up to 103\% at 22.3 $rad/s$.
\end{itemize}
 

\begin{ack}
This research work is funded by the National Key R and D Program of China under Grant No. 2018YFB1702200 and National Natural Science Foundation (Grant No. 62076049). The content of this [report/study/article/publication] does not reflect the official opinion of the National Natural Science Foundation.
\end{ack}

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%\appendix
%\section{A summary of Latin grammar}    % Each appendix must have a short title.
%\section{Some Latin vocabulary}              % Sections and subsections are supported  
                                                                         % in the appendices.
\end{document}
